for a rank-2 symmetric tensor \(g_{\mu \nu }\)  and a scalar \(\phi \) , where the overall constant in front of \(\mathcal {F}\) , which can always be reabsorbed in the flow parameter, was set to 1. The standard metric-dilaton flow equations [1]}
\({\left\lbrace \begin{array}{ll}\mathrm {d}_{\lambda }g_{\mu \nu }=-2R_{\mu \nu }\ ,\\\mathrm {d}_{\lambda }f=-R-\Delta f+\left(\nabla f\right)^2\ ,\\g_{\mu \nu }\left(0\right)=\bar{g}_{\mu \nu }\ ,\\f\left(0\right)=\bar{f}\ ,\end{array}\right.}\) 
